Briefly, we consider the problem when a tangent line to a curve has zero slope. That is, the derivative of the curve is zero at some point.

Such a point is called a **“critical point”** and tells us about extreme values.

**Example.** Consider the function . We plot this out:

Now, we can consider the tangent line at, say, . We draw this tangent line in red:

Observe every point on the line has the property that . In other words, is a **“Global Minimum”** (it’s less than every other if ).

How can we detect such “extrema” (i.e., “maxima” and “minima”)? Well, the tangent line has zero slope. Equivalently: when the point is an extreme point.

Sometimes these “extrema” points are local. For example, plotting when :

So how do we detect extrema?

Well, we take our function , take its derivative, then find points such that .

This is the general routine, but lets leave the motivating post with a few questions…

**Problem:** as we can see with the function , some extrema are local while others are global. How do we

(a) Determine if an extrema is a maximum or minimum?

(b) Determine if the extrema is local or global?

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## About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.

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